Statistics is a mathematical science related to the collection, analysis, interpretation and presentation of data. It has a wide range of applications from the physical and social sciences to the humanities, as well as to business, government, medicine and other various industries. A common goal for most statistical analysis operations is to investigate causality, and in particular to draw conclusions on the effect of various changes to the values of predictors or independent values on a response or dependent variable. If a sample of the total population of data is representative of the population, then inferences and conclusions made from a statistical analysis of the sample can be extended to the entire population as a whole. Furthermore, statistics offers methods to estimate and correct for randomness in the sample and in the data collection procedure. The fundamental mathematical concept employed in understanding such randomness is probability.
A probability distribution is a function that assigns probabilities to given events or propositions. There are several equivalent ways to specify a probability distribution, for example to specify a probability density function or specify a moment. In addition to a probability distribution, a statistics analysis can include the likelihood principle which asserts that all of the information in a sample is contained in a likelihood function.
A typical statistical analysis performed using the likelihood principle uses a binned likelihood function, which is obtained by taking the product of the probabilities for each ordinate data point y, as a function of the fitting parameter(s) α and the abscissa value(s) x as shown in the following equation:L(y|α)=ΠjP(yj,α,xj)  (1)wherein file index j runs over the data set (x,y) of dimension M. The probability P(y|α,x) is a function of two entities: the fitting function f(α,x).), hereafter known as ƒj, and the statistical function that predicts how the gate data will distribute around f, hereafter called the statistics. In the case of Gaussian statistics, for example, the likelihood reads:
                              L          ⁡                      (                          y              ❘              α                        )                          =                              ∏                          j              ⁢                                                                                                                  ⁢                                          ⁢                                    1                                                2                  ⁢                                      πσ                    j                                                                                                                          ⁢                          exp              ⁡                              (                                                      -                                          1                      2                                                        ⁢                                                                                    (                                                                              y                            j                                                    -                                                      f                            j                                                                          )                                            2                                                              σ                      j                      2                                                                      )                                                                        (        2        )            
A standard treatment may involve the maximization of L(y|α) according to the frequenting method, or equivalently the maximization of its logarithm. Finding the maximum of L(y|α) or its logarithm will return a set μ which estimates the unknown set α. Alternatively, the first moments of the likelihood can be used as an estimator for α.
A set of variances D describe the variance of each parameter α. Often D is defined as σ2 in the literature with the result of a measurement given as α±σ. More comprehensively, the set of variances D is related to the diagonal of the covariance matrix V, where the off-diagonal elements carry information regarding correlation between different fitting parameters. A covariant matrix can be defined as:
                                          V            ik                    =                      -                                                            ∂                  2                                ⁢                L                                                              ∂                                      α                    i                                                  ⁢                                  ∂                                      α                    k                                                                                      ⁢                                  ⁢        with                            (        3        )                                          1                      D            i                          =                  V          ii                                    (        4        )            Alternatively, one can derive the covariance matrix from second moments:
                              1                      V            ik                          =                                            ∫                                                ⅆ                                      α                    i                                                  ⁢                                  ⅆ                                      α                    k                                                  ⁢                                  L                  ⁡                                      (                                          y                      ❘                      α                                        )                                                  ⁢                                  α                  i                                ⁢                                  α                  k                                                                    ∫                                                ⅆ                                      α                    i                                                  ⁢                                  ⅆ                                      α                    k                                                  ⁢                                  L                  ⁡                                      (                                          y                      ❘                      α                                        )                                                                                -                                    μ              i                        ⁢                          μ              k                                                          (        5        )            
If fluctuations associated with data (x,y) are distributed according to a Gaussian probability distribution, then the statistics for the fit of the data are Gaussian. Likewise, if the fluctuations are distributed according to a Poissonian, multiracial or a rarer type of probability distribution, the statistics are described by the given distribution.
Regardless of matter which distribution is used, a goodness-of-fit parameter specifies how faithfully data are reproduced by a given fitting function. For example, in the case of using Gaussian statistics, the familiar χ2 defined as:χ2=−2 ln Lmax   (6)provides information as to how well a Gaussian distribution fits the data.
Using a Bayesian framework, a fitting procedure is similar to the frequenting method, but the likelihood is related to the posterior probability P(α|y) via Bayes' theorem:
                              P          ⁡                      (                          α              ❘              y                        )                          =                                            L              ⁡                              (                                  y                  ❘                  α                                )                                      ⁢                          P              ⁡                              (                α                )                                                          P            ⁡                          (              y              )                                                          (        7        )            wherein the prior probability P(α) includes any prior information on the fit. Thereafter a similar procedure as used in the frequenting method, i.e. maximization, estimation of the uncertainty and estimation of goodness-to-fit, is applied to P(α|y). In the alternative, an estimation of goodness-to-fit is evaluated using the integral of the posterior probability P(α|y).
Whichever method is used, frequentist or Bayesian, it is important to note that in the vast majority of statistical analysis problems the difference in results obtained using either method is too small to be of practical consequence. However, there is a fundamental limitation in the methods presently used to statistically analyze data. This limitation is the inability of present methods to effectively identify biased data using current goodness-of-fit parameters.
This limitation is even greater when statistical analysis is used to combine several sets of data with all the sets measuring the same set of parameter(s) α. For example, data from clinical tests are partially or completing overlapping, with the main problem being in determining the exact weight to be given to each test in order to extract a more precise value of the parameter(s) being studied. Thus, current methods give data with significant bias the same weight as data with essentially no or very little bias. Therefore, there is a need for on improved method for information analysis.